Place a chicken breast on a cutting board, skin side up. Holding knife parallel to board, cut through breast along a long side, stopping about ½" before you get all the way through. Open breast up like a book, with the connected side acting as the spine. Place butterflied breast between 2 pieces of plastic wrap and gently pound until as thin as possible without tearing meat, ideally to about ¼" thick and about 10" in diameter. Repeat with remaining chicken breast.
Place eggs, breadcrumbs, and flour in separate shallow bowls (cake pans or pie plates work great). Season each generously with salt and pepper. Working with 1 cutlet at a time, season lightly with salt and pepper and dredge in flour, shaking off excess. Dip in eggs, letting excess drip back into bowl, then coat in breadcrumbs, shaking off excess. Place on a rimmed baking sheet.
Heat oil in a large skillet over medium-high until very hot (an instant-read thermometer should read 450°–475° and oil should be just beginning to smoke). Working with 1 cutlet at a time, very carefully lower along the side of skillet closest to you and let slide into oil so it lies flat. Swirl oil in skillet carefully and cook just until cutlet is browned underneath, about 1 minute. Using tongs, carefully turn cutlet and cook until golden brown on other side, about 1 minute (it does not need to be fully cooked through as it will cook more under broiler). Transfer to a wire rack set inside a foil-lined rimmed baking sheet.
Heat broiler. Top each cutlet with ½ cup warm Monte’s Tomato Sauce, spreading to edges. Evenly scatter ¼ cup Parmesan & slices of fresh mozzarella over. Broil until cheese is aggressively bubbling, about 2 minutes.